The limit of the m-norms of a class of symmetric matrices and its applications

TL;DR

This paper derives a Weyl-like formula for a special class of symmetric matrices, applies it to compute integrals involving the gamma function, and investigates their properties related to Hadamard matrices.

Contribution

It introduces a new formula for symmetric matrices similar to Weyl's criterion and explores its applications in integral calculation and matrix classification.

Findings

Derived a Weyl-like formula for a class of symmetric matrices

Provided a new method to compute the integral of ln Gamma(x) on [0,1]

Showed that certain matrices are not Hadamard matrices

Abstract

We consider a special symmetric matrix and obtain a similar formula as the one obtained by Weyl's criterion. Some applications of the formula are given, where we give a new way to calculate the integral of $\ln\Gamma(x)$ on $[0,1]$, and we claim that one class of matrices are not Hadamard matrices.